Flex plate uses a automatic transmission. A flywheel uses a manual transmission.
However, this is a bit misleading if your referring to chevy flywheels, as the 400 small block used a 168 tooth flywheel and flexplate, but it has balance weights on it because the 400 was the only generation 1 small block that was externally balanced. So the harmonic damper and flywheel were specially for the 400, if a non balanced balancer or flexplate is used on a 400 it`ll destroy the thrust surface. If the 400 pieces are used on a smaller engine such as a 350 or any other small block, it knocks it out of balance so bad you`ll have a real shaker on your hands. When the sbc went to a 1 piece seal crank in 1986, things changed again. To use a 1 piece rear seal the flange had to be made completely round so the crank had to be balanced on the flywheel. But chevy got it right by making it where the old 1 and 2 piece seal flywheels / flex plates wouldn`t interchange. So the 1 piece seal engines are unique, in the front they are internally balanced and in the rear they are externally balanced.